Loop homotopy of $6$-manifolds over $4$-manifolds

Abstract

Let M be the 6-manifold M as the total space of the sphere bundle of a rank 3 vector bundle over a simply connected closed 4-manifold. We show that after looping M is homotopy equivalent to a product of loops on spheres in general. This particularly implies the cohomology rigidity property of M after looping. Furthermore, passing to the rational homotopy, we show that such M is Koszul in the sense of Berglund.

Sphere bundles over $4$-manifolds are trivial after looping

“…The topology of circle bundles over 4-manifolds has been studied by [DL05], from which Beben and Theriault [BT14] have determined their loop homotopy types. The homotopy of 2-sphere bundles was studied by the author recently [Hua22]. Our main theorem below determines the loop homotopy types of sphere bundles over simply connected 4-manifolds for almost all other cases.…”

“…with n = 2 or 3. Partial results were obtained by the author [Hua22] for n = 2 and by Theriault and the author [HT22] for n = 3. Based on the results there, it can be realized that these two cases are quite different from the other cases when n ≥ 4.…”